Definition of Taylor series:

f(x)=f(a)+f′(a)(x−a)+f′′(a)(x−a)22!+⋯+f(n−1)(a)(x−a)n−1(n−1)!+Rn

Rn=f(n)(ξ)(x−a)nn! where a≤ξ≤x, ( Lagrangue’s form )

Rn=f(n)(ξ)(x−ξ)n−1(x−a)(n−1)! where a≤ξ≤x, ( Cauch’s form )

This result holds if f(x) has continuous derivatives of order n at last. If limn→+∞Rn=0, the infinite series obtained is called Taylor series for f(x) about x=a. If a=0 the series is often called a Maclaurin series.